NESTA: A Fast and Accurate First-order Method for Sparse Recovery

TL;DR

This paper introduces NESTA, a fast, accurate, and flexible first-order optimization algorithm inspired by Nesterov's smoothing technique, optimized for large-scale sparse recovery and compressed sensing problems.

Contribution

The paper presents NESTA, a novel first-order method that improves convergence and robustness for sparse recovery, with broad applicability and minimal parameter tuning.

Findings

NESTA outperforms existing methods in accuracy and speed.

It is effective for large-scale compressed sensing problems.

The algorithm is versatile, handling various convex optimization tasks.

Abstract

Accurate signal recovery or image reconstruction from indirect and possibly undersampled data is a topic of considerable interest; for example, the literature in the recent field of compressed sensing is already quite immense. Inspired by recent breakthroughs in the development of novel first-order methods in convex optimization, most notably Nesterov's smoothing technique, this paper introduces a fast and accurate algorithm for solving common recovery problems in signal processing. In the spirit of Nesterov's work, one of the key ideas of this algorithm is a subtle averaging of sequences of iterates, which has been shown to improve the convergence properties of standard gradient-descent algorithms. This paper demonstrates that this approach is ideally suited for solving large-scale compressed sensing reconstruction problems as 1) it is computationally efficient, 2) it is accurate and returns solutions with several correct digits, 3) it is flexible and amenable to many kinds of reconstruction problems, and 4) it is robust in the sense that its excellent performance across a wide range of problems does not depend on the fine tuning of several parameters. Comprehensive numerical experiments on realistic signals exhibiting a large dynamic range show that this algorithm compares favorably with recently proposed state-of-the-art methods. We also apply the algorithm to solve other problems for which there are fewer alternatives, such as total-variation minimization, and convex programs seeking to minimize the l1 norm of Wx under constraints, in which W is not diagonal.